Taking the long way around

The Chicago Art Museum's Renaissance display consists of four hallways bordered around a square courtyard. A single guard is assigned to patrol the four hallways. When the guard starts working, he begins in one of the corners and walks clockwise. When he arrives at a subsequent corner he flips two coins. If both coins are heads, he changes the direction he is walking. Otherwise, he continues in the same direction. Let $$E$$ be the expected number of lengths of hallway that he walks before he first returns to his starting corner. Let $$p$$ be the probability that he walks strictly more than $$E$$ lengths of hallway before returning to his starting corner. $$p$$ can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

Details and assumptions

The guard may walk in the same hallway more than one time. Each time he walks in it counts as one length of hallway.

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